PROCESS CAPABILITY ANALYSES BASED ON RANGE WITH TRIANGULAR FUZZY
NUMBERS
S. SELVA ARUL PANDIAN1 & P. PUTHIYANAYAGAM
2
1Assistant Professor, Department of Mathematics, K.S.R College of Engineering, Tiruchengode, India
2Professor and Controller of Examination, Department of Statistics, Subbalakshmi Lakshmipathy College of Science,
Madurai, India
ABSTRACT
Process Capability Indices (PCI) is an effective and excellent method of measuring product quality and determine
whether the production process produce product within the specified limits. Cp , Cpk , Cpm and Cpmk are most used
traditional process capability indices. In this paper, the fuzzy set theory concept is used to extent the process capability
analyses by using the range (estimate of σ) and fuzzy midrange transform techniques to get more information and
flexibility of the PCA system. The fuzzy tolerance control limits for , and are developed and compared
their performances with a numerical example. The result shows that the , and control limits are the
evidence of improvement in the process performance.
KEYWORDS: Fuzzy Range Control Limits, Fuzzy Tolerance Limits, Process Performance Fuzzy Limits, Process Fuzzy
Control Limits, Triangular Fuzzy Limits
INTRODUCTION
Process capability indices are widely used to check whether the production process performance is within the
customer’s requirement. A process meeting customer requirement is called “capable”. A process capability (PCI) is a
process characteristic relative to specifications. The setting and communication becomes vey simpler and easier by using
process capability indices to express process capability between manufacturers ad customers. The use of these indices
provides a unit less language for evaluating not only the actual performance of production processes, but the potential
performance as well.
The indices are intended to provide a concise summary of importance that is readily usable. Engineers,
manufacturers and suppliers can communicate with this unit less language in an effective manner to maintain high process
capabilities and enable cost savings. The capability of a process and effectiveness of control charts are directly related.
PCI’s have become very popular in assessing the capability of manufacturing process in practice during the past decade.
Process capability can be broadly defined as the ability of a process to meet customers’ expectations which are defined as
specification limits.
The PCA’s compares the output of a process to the specification limits by using the process capability indices
(PCI). In the review of the literature, Cp , Cpk , Cpm and Cpmk are some of indices used to identify. In this paper, the study
is structured in the following order. The literature reviews of various Process capability Indices were discussed in the first
section. Secondly, the approximate tolerance limits for , and based on range are constructed by using
chi-square distributions by Patnaik approximation. Thirdly, the approximate tolerance limits for based on range is
transformed to fuzzy Triangular number tolerance limits for , and based on range by using α-level fuzzy
International Journal of Applied Mathematics &
Statistical Sciences (IJAMSS)
ISSN 2319-3972
Vol. 2, Issue 1, Feb 2013, 1-12
© IASET
2 S. Selva Arul Pandian & P. Puthiyanayagam
midrange transform technique. Fourthly, the fuzzy estimations of , and chart based on range by using α-
cuts are constructed. Next, the applications to understanding the fuzzy estimations of , and chart based on
range by using α-cuts are given and finally, the conclusions are presented.
LITERATURE REVIEWS
There are different indices that are given in literature review. In this case the quality characteristics X and the
corresponding random sample (x1, x2,x3,……xn) are normal, in fact . Let LSL and USL denotes the lower
and upper specification limits, , the midpoint of tolerance interval (LSL, USL), t the target value for µ,
which we assume that t = m.
Kane (1986) was suggested the simplest and process potential index defined as
(1)
This index is a ratio of tolerance region to process region. It is clear that this method consider the variation of
process. 6 in denominator of above fraction are based on assumption of approximately normal of data. It will react to
change in process dispersion but not change of process location.
Sullivan (1984) has been suggested a new process index CPK in order to reflect departure from the target value as
well as change in the process variation, is given by
(2)
Chan, Chen and Springe (1988) given another new process index CPM, in order to be a sign of departure from the
target value as well as change in the process variation, defined and is given by
(3)
6σ, 3σ in denominator of above fractions are based on assumption of approximately normal of data. In the other
words main constraints in above indices is its normality assumptions. Departure from the target value carry more weight
with the other well known capability indices is defined by
(4)
Cpm and Cpmk react more both in dispersion and location than Cpk. Cpmk is more sensitive than Cpm to deviations
from the target value T.
Kerstin Vannman (1995), has been given the unified approach. By varying the parameters of this class, we can
find indices with different desirable properties. The proposed new, indices depend on two non-negative parameters, u and
v, as
(5)
It is easy to verify that:
Process Capability Analyses Based on Range with Triangular Fuzzy Numbers 3
Form the study of , large value of u and v will make the index more sensitive to departure from
the target value. A slight modification gives general index class which includes
(6)
The five are equal when µ = T =M, but differ when .
Carr (1991), The other approach is defined to use non conforming ratios as an index for capability process for the first time
(7)
This approach based on the stepwise loss function as below:
Clements (1989), in an influential paper, suggested that “6σ” be replace by the length of the interval between the
upper and lower 0.135 percentage points of the distribution of X and defined
(8)
Greenwich and Jahr-Schaffrath (1995) defined the index Cpp which provides an uncontaminated separation
between information concerning process accuracy and process precision as follows
(9)
Where the inaccuracy index and
Imprecision index and .
Bernardo and Irony (1996), The bayes capability index CB(D) given by
(10)
A Bayesian index is proposed to evaluate process capability which within a decision –theoretical framework,
directly assesses the proportion of future work may be expected to lie outside the tolerance limits. Hsin-Lin Kuo (2010)
extended the capability indices by getting approximate tolerance limits for Cp capability chart based on Range
Approximate Tolerance Limits for , & Based on Range using χ2 Distribution
Let X1, X2, X3,……, Xn be a random sample of size n drawn from a normal population with mean µ and standard
deviation σ. The mean of the m ranges will be denoted by and the range of a single sample of size n is denoted by
R1,n.
When E ( ) = σd2 and Var(R) = σ2
, the mean and variance of are given by
and
4 S. Selva Arul Pandian & P. Puthiyanayagam
, respectively
Then is the unbiased estimator of σ, where and are constants. According to Patnaik, it has been
shown that is approximately distributed as That is
and
Where denotes a chi-square distribution with degrees of freedom, and c and are constants which are
functions of the first two moments of the range variable, given by
Using this relations, the values of c and can be easily obtained for any n and m. Assume that the process the
measurement follows N (µ, σ2), the normal distribution, the , and index are given below
(11)
(12)
(13)
Apply a simple approximation procedure based on range we can obtain the tolerance limits of the estimator of .
The 100(1 – α) approximate tolerance limits for together with R charts
Where and is the upper quantile of the chi-square
distribution with v degrees of freedom. So, the 100(1 – α) approximate tolerance limits for , and based
on range is given below
Upper Tolerance Limit Center Line Lower Tolerance Limit
(14)
Fuzzy Transformation Techniques and α - Level Fuzzy Mid – Range
In this study, α- level fuzzy mid – range transformation techniques used for the construction of approximate
tolerance limits for based on range to fuzzy triangular number control chart. The α- level fuzzy mid – range is
Process Capability Analyses Based on Range with Triangular Fuzzy Numbers 5
defined as the midpoint of the ends of the α- cuts. An α- level cut, denoted by Aα , is a nonfuzzy set that comprises ale
element whose membership is greater than or equal to α. If aα and b
α are the end points of A
α , then
(15)
α- level fuzzy mid – range of the sample is given by,
(16)
Fuzzy Approximate Tolerance Limits for , and Based on Range
α –cut Fuzzy tolerance limits for based on Range
Upper tolerance limit:
Center line: (17)
Lower tolerance limit:
Where (18)
(19)
α- level Fuzzy midrange tolerance limits for based on Range
Upper tolerance limit:
Center line: (20)
Lower tolerance limit:
The condition of the process control can be defined by
(21)
α –Cut Fuzzy Tolerance Limits for Based on Range
α –cut fuzzy tolerance limits for based on Range can be calculated as follows.
Upper tolerance limit:
Center line: (22)
Lower tolerance limit:
6 S. Selva Arul Pandian & P. Puthiyanayagam
Where (23)
(24)
α- level Fuzzy midrange tolerance limits for based on Range
Upper tolerance limit:
Center line: (25)
Lower tolerance limit:
The condition of the process control can be defined by
(26)
α –cut Fuzzy tolerance limits for based on Range
Upper tolerance limit:
Center line: (27)
Lower tolerance limit:
Where (28)
(29)
α- level Fuzzy midrange tolerance limits for based on Range
Upper tolerance limit:
Center line: (30)
Lower tolerance limit:
The condition of the process control can be defined by
(31)
Process Capability Analyses Based on Range with Triangular Fuzzy Numbers 7
Table 1: The Triangular Fuzzy Measurement Values & the Fuzzy Ranges
Sa
mp
le Xa Xb Xc
1 2 3 4 5 1 2 3 4 5 1 2 3 4 5
1 5.71 5.5 5.43 5.2 5.51 5.73 5.57 5.45 5.25 5.53 5.75 5.6 5.46 5.27 5.55
2 5.41 5.52 5.25 5.51 5.65 5.43 5.57 5.29 5.53 5.69 5.44 5.58 5.3 5.56 5.7
3 5.25 5.51 5 5.2 5.31 5.29 5.53 5.13 5.25 5.33 5.32 5.54 5.17 5.28 5.37
4 5.42 5.26 5.42 5.49 5.6 5.51 5.31 5.44 5.55 5.64 5.62 5.38 5.46 5.58 5.71
5 5.19 5.18 5.25 5.21 5.52 5.3 5.2 5.28 5.27 5.57 5.32 5.31 5.32 5.3 5.6
6 5.36 5.31 5.18 5.38 5.26 5.42 5.4 5.23 5.48 5.33 5.63 5.55 5.31 5.47 5.42
7 5.26 5.53 5.41 5.28 5.19 5.27 5.57 5.46 5.29 5.26 5.31 5.62 5.49 5.32 5.3
8 5.43 5.28 5.44 5.5 5.58 5.52 5.33 5.47 5.56 5.62 5.63 5.4 5.5 5.6 5.69
9 5.69 5.45 5.32 5.19 5.45 5.72 5.49 5.35 5.22 5.48 5.75 5.56 5.41 5.28 5.51
10 5.31 5.26 5.14 5.4 5.31 5.35 5.29 5.21 5.43 5.36 5.42 5.32 5.26 5.46 5.38
11 5.28 5.5 5.41 5.31 5.35 5.32 5.56 5.46 5.34 5.41 5.36 5.62 5.48 5.38 5.44
12 5.43 5.22 5.15 5.34 5.48 5.46 5.26 5.18 5.36 5.52 5.48 5.28 5.21 5.41 5.56
13 5.46 5.35 5.35 5.22 5.28 5.52 5.39 5.42 5.28 5.32 5.56 5.43 5.48 5.34 5.36
14 5.41 5.36 5.52 5.51 5.38 5.44 5.4 5.56 5.54 5.42 5.48 5.43 5.62 5.62 5.46
15 5.62 5.48 5.42 5.18 5.41 5.64 5.55 5.46 5.23 5.46 5.68 5.59 5.49 5.26 5.49
Applications of α - Level Fuzzy Midrange Tolerance Limits for , and
An application was given by Sentruk and Erginel [9] made on controlling piston inner diameters in compressors.
The same data have been considered with the first fifteen samples each with size 5 (the total measurements is 5 X 15 = 75)
were taken from the production process by different operators. Quality experts evaluated each value by a fuzzy number
because the variability of a measurement system includes operators and gauge variability. For example, the numeric
observed value 5.73 can be measured by various operators between 5.71 and 5.76.The upper and lower specification limits
of the process are defined as approximately 5.7 and approximately 5.1, respectively. These measurements are converted
into triangular fuzzy numbers and given in Table 1. Fuzzy control limits are calculated according to the procedures given
in the previous sections.
From the process data, If we assume that mean ( and is the estimate of µ and σ.
We obtain the following result:
0.332 , c = 2.3367 v = 54.59
Target value t = 5.4 Mean ( = 5.4163
8 S. Selva Arul Pandian & P. Puthiyanayagam
Table 2: α - Level Fuzzy Midrange Tolerance Limits for Based on Range
Sample
No.
1 0.3497 0.3134 0.2811 0.3141 Out of control
2 0.4497 0.3838 0.3567 0.3906 Out of control
3 0.2341 0.3993 0.4945 0.3871 Out of control
4 0.5975 0.4934 0.3524 0.4869 Out of control
5 0.3877 0.4694 0.6978 0.4951 Out of control
6 0.7676 0.8436 0.5427 0.7776 In control
7 0.5336 0.6753 0.7075 0.6561 In control
8 0.6564 0.5347 0.3636 0.5261 Out of control
9 0.4342 0.3846 0.3266 0.3831 Out of control
10 0.5487 0.8035 1.0389 0.8001 In control
11 0.9515 0.9110 0.7276 0.8860 In control
12 0.5263 0.5838 0.6380 0.5832 In control
13 0.7495 0.9239 0.9374 0.8958 In control
14 1.2793 1.1049 0.7264 1.0691 In control
15 0.4899 0.4387 0.3655 0.4349 Out of control
α - Level Fuzzy Midrange Tolerance Limits for Based on Range
Upper tolerance limit
Center line: (32)
Lower tolerance limit
have been calculated for all the 15 samples with respect to different operators by using equations (16)
and are given in the table 2. As shown in the table 2, the process capability index decreased and increased in samples like
1, 2, 3, 4, 5, 8, 9 and 15 which indicating the process is out of control and need of investigations. Out of 15 samples 8
sample points lies outside of the process control.
Table 3: α - Level Fuzzy Midrange Tolerance Limits for Based on Range
Sample
No.
1 0.4345 0.431 0.4136 0.4286 Out of control
2 0.5408 0.5001 0.4821 0.5041 Out of control
3 0.3796 0.5102 0.5832 0.5002 Out of control
4 0.6621 0.5952 0.4843 0.5875 In control
5 0.5112 0.5672 0.7552 0.5903 In control
6 0.7496 0.9003 0.6362 0.8277 In control
7 0.6235 0.732 0.7256 0.7119 In control
8 0.7303 0.6257 0.4854 0.6194 In control
9 0.4632 0.4522 0.4418 0.4523 Out of control
10 0.6208 0.8413 1.0900 0.8462 In control
11 1.0078 0.9547 0.7999 0.9369 In control
12 0.6213 0.6551 0.6625 0.6505 In control
13 0.8092 0.9604 0.9949 0.9400 In control
14 1.2880 1.0042 0.6811 0.9974 In control
15 0.5251 0.5293 0.4822 0.5203 Out of control
Process Capability Analyses Based on Range with Triangular Fuzzy Numbers 9
α - Level Fuzzy Midrange Tolerance Limits for Based on Range
α- level Fuzzy midrange tolerance limits for based on Range calculated as
Upper tolerance limit
Center line: (33)
Lower tolerance limit
have been calculated for all the 15 samples with respect to different operators by using equations (16)
and are given in the table 3. As shown in the table 3, the process is in control but the process capability index decreased
and increased in samples like 1, 2,3, 9 and 15 which indicating the process is out of control and need of investigations.
Out of 15 sample only 5 samples lies outside of the process control. The α - level Fuzzy midrange tolerance limits for
based on Range can be used to control the process.
Table 4: α - Level Fuzzy Midrange Tolerance Limits for Based on Range
Sample
No.
1 0.5064 0.4502 0.3875 0.4491 Out of control
2 0.6633 0.6398 0.5731 0.6323 In control
3 0.1949 0.3504 0.4588 0.3421 Out of control
4 0.7460 0.7737 0.5690 0.7331 In control
5 0.2897 0.4235 0.6797 0.4449 Out of control
6 0.4947 0.8163 0.7974 0.7567 In control
7 0.4863 0.6588 0.7450 0.6437 In control
8 0.8423 0.8342 0.5883 0.7926 In control
9 0.4941 0.5305 0.4810 0.5155 Out of control
10 0.3807 0.6394 0.9737 0.6526 In control
11 0.9070 1.012 0.9492 0.9827 In control
12 0.4639 0.559 0.6360 0.5558 In control
13 0.6258 0.9155 1.1077 0.8985 In control
14 1.4426 1.2453 0.9581 1.2295 Out of control
15 0.5636 0.6493 0.5875 0.6235 Out of control
α - Level Fuzzy Midrange Tolerance Limits for Based on Range
α- level Fuzzy midrange tolerance limits for based on Range calculated as
Upper tolerance limit
Center line: (34)
Lower tolerance limit
have been calculated for all the 15 samples with respect to different operators by using equations (16)
and are given in the table 4.
As shown in the table4, the process is in control but the process capability index decreased and increased in
samples like 1, 3, 5, 9, 14 and 15 which indicating the process is out of control and need of investigations. Out of 15
10 S. Selva Arul Pandian & P. Puthiyanayagam
sample only 6 samples lies outside of the process control. The α - level Fuzzy midrange tolerance limits for based on
Range can be used to control the process.
CONCLUSIONS
In this paper, it is proposed a new methodology of constructing the fuzzy tolerance control limits for ,
and in fuzzy set theory. Process capability analysis is an important tool to improve the process performance. The
fuzzy tolerance control limits for , and , improve the process control as compared with traditional control
limits. By comparing the fuzzy control limits of , and , the fuzzy control limits shows that the out of
control points is fewer than the and .
The fuzzy control limits is the enhanced process limit among the other fuzzy control limits in 3σ and also it
shows the substantiation in improvement in the process performance. Fuzzy set theory is also befitted method for
improving the process performance in process capability analysis. In the future, the performance of the process is assessing
by using other transformation techniques in fuzzy logic.
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Process Capability Analyses Based on Range with Triangular Fuzzy Numbers 11
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